OCR FM1 AS (Further Mechanics 1 AS) 2018 March

1 A particle \(P\) of mass 2.4 kg is attached to one end of a light inextensible string of length 1.4 m . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal table. \(P\) moves on the table at constant speed along a circular path with \(O\) at its centre. The magnitude of the tension in the string is 21 N .

1. (a) Find the magnitude of the acceleration of \(P\).
   (b) State the direction of the acceleration of \(P\).
2. Find the speed of \(P\).
3. Find the time taken for \(P\) to complete a single revolution.

2 A pump is pumping still water from the base of a well at a constant rate of 300 kg per minute. The well is 4.5 m deep and water is released from the pump at ground level in a horizontal jet with a speed of \(6.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring any energy losses due to resistance, calculate the power generated by the pump.

3 A student is investigating fluid flowing through a pipe.
In her first model she assumes a relationship of the form \(P = S \rho ^ { \alpha } g ^ { \beta } h ^ { \gamma }\) where \(\rho\) is the density of the fluid, \(h\) is the length of the pipe, \(P\) is the pressure difference between the ends of the pipe, \(g\) is the acceleration due to gravity and \(S\) is a dimensionless constant. You are given that \(\rho\) is measured in \(\mathrm { kg } \mathrm { m } ^ { - 3 }\).

1. Use the fact that pressure is force per unit area to show that \([ P ] = \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 }\).
2. Find the values of \(\alpha , \beta\) and \(\gamma\). The density of the fluid the student is using is \(540 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). In her experiment she finds that when the length of the pipe is 1.40 m the pressure difference between the ends of the pipe is \(3.25 \mathrm { Nm } ^ { - 2 }\).
3. Find the length of the pipe for which her first model would predict a pressure difference between the ends of the pipe of \(4.65 \mathrm { Nm } ^ { - 2 }\). In an alternative model the student suggests a modified relationship of the form \(P = S \rho ^ { \alpha } g ^ { \beta } h ^ { \gamma } + \frac { 1 } { 2 } h v ^ { 2 }\), where \(v\) is the average velocity of the fluid in the pipe.
4. Use dimensional analysis to assess the validity of her alternative model.

4 A car has a mass of 850 kg and its engine can generate a maximum power of 35 kW . The total resistance to motion of the car is modelled as \(k v \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. When the car is moving in a straight line on a straight horizontal road, the greatest constant speed that it can attain is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 56\).
2. Find the greatest possible acceleration of the car on the road at an instant when it is moving with a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A trailer of mass 240 kg is attached to the car by means of a light inextensible tow bar which is parallel to the surface of the road. The resistance to motion of the trailer is modelled as a constant force of magnitude 350 N . The car and trailer move on the horizontal road. At a certain instant the car's engine is working at a rate of 30 kW and the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. (a) Find the speed of the car at this instant.
   (b) Find the magnitude of the tension in the tow bar at this instant. The car and trailer now move in a straight line on a straight road inclined at \(8 ^ { \circ }\) to the horizontal.
4. Find the difference between their greatest possible constant speed travelling up the slope and their greatest possible constant speed travelling down the slope.

5 Two particles \(A\) and \(B\) are on a smooth horizontal floor with \(B\) between \(A\) and a vertical wall. The masses of \(A\) and \(B\) are 4 kg and 11 kg respectively. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
\includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-3_209_803_1658_630}

1. Show that immediately after the collision the speed of \(B\) is \(\frac { 4 } { 15 } u ( 1 + e )\). After the collision between \(A\) and \(B\) the direction of motion of \(A\) is reversed. \(B\) subsequently collides directly with the vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 } e\).
2. Given that there is a second collision between \(A\) and \(B\), find the range of possible values of \(e\).

6 A fairground game involves a player kicking a ball, \(B\), from rest so as to project it with a horizontal velocity of magnitude \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is attached to one end of a light rod of length \(l \mathrm {~m}\). The other end of the rod is smoothly hinged at a fixed point \(O\) so that \(B\) can only move in the vertical plane which contains \(O\), a fixed barrier and a bell which is fixed \(l \mathrm {~m}\) vertically above \(O\). Initially \(B\) is vertically below \(O\). The barrier is positioned so that when \(B\) collides directly with the barrier, \(O B\) makes an angle \(\theta\) with the downwards vertical through \(O\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-4_643_659_584_724} The coefficient of restitution between \(B\) and the barrier is \(e . B\) rebounds from the barrier, passes through its original position and continues on a circular path towards the bell. The bell will only ring if the ball strikes it with a speed of at least \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The player wins the game if the player causes the bell to ring having kicked \(B\) so that it first collides with the barrier. You may assume that \(B\) and the bell are small and that the barrier has negligible thickness. Show that, whatever the position of the barrier, the player cannot win the game if \(u ^ { 2 } < 4 g l + \frac { V ^ { 2 } } { e ^ { 2 } }\). \section*{END OF QUESTION PAPER}